VIII. Rubber Elasticity [B.Erman, J.E.Mark, Structure and properties of rubberlike networks] Using various chemistry, one can chemically crosslink polymer chains. With sufficient cross-linking, the polymer system forms a network. Network defects will be: -Dangling ends -Loops. Main assumptions of the classical theories: Phantom flexible chains; Only intra-molecular effects, no inter-molecular interactions Important characteristics of the cross-linked network: Strands, the averaged molecular weight between cross-links M C ; Junctions, their functionality ; The total number of network chains, ; The number of junctions, ; The cycle rank (how many chains to cut in order to reduce the network to a tree with no closed cycles) These 5 parameters are not independent: The number of chain ends 2 is equal to the number of functional groups : =2. 2 N The other relations are not so obvious: ; A 1 V0 M C Here V 0 is the volume of network in the state of formation, is the corresponding density. In imperfect network, the parameters should be calculated accordingly.
Elementary statistical theory for idealized networks
According to the thermodynamic expression: Relationship between Stress and Strain where t is the stress along the t-th coordinate direction, t is the ratio of the final length to the reference length along that direction. Let s consider a prismatic block of a network. The equality of the volume before and after the application of the stresses is the result of assuming incompressibility. The deformation ratio: t =L t /L 0t. Thus three deformation ratios: 1 2 3 =V/V 0. The deformation ratio t relative to the initial dimensions t =L t /L it. Assuming the network is 3 V / V 1/ A 1 el t V t t Block of the reference volume Vo T, V (14) The initial state before deformation. V may differ from Vo due to T or solvent isotropic in the undisorted state: t 0 t 3 2 3 2 1 Ael i FkT i One can rewrite eq.14, using eq.11: t V t t (15) i1 2 i t V i1 t The final dimension under stress. V~const, no solvent leaves or enter network. Specification of t for any deformation gives the desired equation of state. Setting F=/2 yields the affine limit, F=/2 the phantom limit. A knowledge of the volume fraction of polymer for any molecular interpretation: Fraction during network formation: v2 c Vd /V 0 Fraction during the stress strain experiment: v V d / V V d is the volume of the dry network, V i is the volume of the network + solvent at the start of the stressstrain experiment. Thus 1/ 3 1/ 3 V / V v 0 v2c / 2 2 i
Statistical Theory for Real Networks The two models described above are the simplest models. The affine network model assumes junctions are fixed and translate affinely with macroscopic strain. No assumption is made with regard to a chain between junctions. In the phantom model the junctions reflect the full mobility of the chain. The chains are assumed to be phantom. Thus there are two extremes. A real network is expected to exhibit properties that fall between those two extremes: junctions fluctuations do occur but they are limited. Limitations appear due to entanglements. The degree of entanglement in a network is proportional to the number of chains sharing the volume occupied by the given chain. Stretching increases the space available to a chain along the direction of the stretch. The chain has more freedom to fluctuate. The same effect can appear during swelling because the separation between chains increases. As a result the modulus should decrease with stretching and swelling. This is supported by experimental results.
Constrained junction model (Flory 1977) The model assumes that the fluctuations of junctions are affected by interpenetration of chains. 4 2 3 / 2 The average number of junctions within this domain: r 3 0 V0 For typical network ~25-100. A quantitative measure of the strength of the constraints is given by the ratio: 2 R s In the phantom limit the constraints are inoperative, 2 ph 2 s ; 0 In the affine limit the constraints are infinitely strong, 0 2 s 0; 0 0 Two types of forces act on a junction: The restoring action of the phantom network pulls the junction toward its mean position; Constraining effects of the other junctions, this force is directed toward a center of constraints for the chosen junction. The elastic free energy is a sum of these two contributions: A A A el ph c
Uniaxial Extension
Rubber reinforcements At significant extensions the rubber modulus increases significantly. There are two primary reasons: Non-Gaussian deformation of chains, approaching limit of the chain expansion, Stress-induced crystallization For many applications usual rubber modulus is too weak. So, various ways of rubber reinforcements are employed. We will consider two major approaches: Filled Elastomers Networks with Multi-modal chain length distribution Filled Elastomers One of the traditional ways of rubber reinforcements is addition of fillers. This way is widely used in practical applications. The two most important are addition of carbon black to natural rubber and some synthetic elastomers; addition of silica to silicone rubbers. The mechanism of the reinforcement on a molecular level is only poorly understood. One of the important mechanisms: chain may adsorb strongly onto the particle surfaces, increasing effectively degree of cross-linking.
Increase in the modulus is due to the limited extensibility of the very short chains. Typical plots of nominal stress against elongation for a swollen bimodal PDMS networks. Long chains M c ~18000, and very short chains M c ~1100 (), 660 () and 220 (). Numbers show mol% of short chains. The area under each curve represents the rupture energy E r. Increase in the upturn in [f*] of bimodal networks can be obtained: By increasing the number of short chains; properties improve up to a short-chain concentration ~95 mol%, at higher concentrations the network becomes brittle; By decreasing the chain length. The Money-Rivlin representation of similar results: Unswollen PDMS networks in elongation at T=-45C. M c =18500 and 220. Increase of elongation (open circles) agrees with decrease of elongation (closed circle).
Concluding Remarks Rubber elasticity is based on purely entropic force, and this provides rather weak elastic modulus, but high extensibility. The modulus is defined by the molecular weight between crosslinking, but might be also affected by chain entanglement at low crosslinking density. Constrained junction model provides reasonable description for the main properties of rubber elasticity. Although rubber elastic force increases with temperature (due to its entropic nature), correction to the elastic force depends on whether expanded or compact state of the chain has lower conformational energy. There are different ways of rubber reinforcement. Traditional way based on adding nanoparticles usually leads to hysteresis effects in mechanical properties under deformation.
Conclusions to the entire class materials We discussed structure of the chain, including molecular weight, PDI and configurations (isomeric states), and experimental methods of their analysis. We discussed chain statistics and conformations, freely joined chain (Gaussian chain) as the basis of polymer chain description. Thermodynamics of polymer solutions and blends based on analysis of Gibbs free energy. Taking enthalpic and entropic contributions into account, and introducing effective parameter provides description of the phase diagram. Entropy of mixing is very low in the case of polymers. 1 2 1 1 B 1 ( ln kt kt N N 1 ln2 1 2) ln1 ln 11 1 1 eff A H S N N T G m 1 1 2 1 2 Crystallization and melting discussion is also based on analysis of Gibbs free energy, and surface tension contribution. In contrast to small molecules, polymer crystallization and melting occur over very broad temperature range. G f G G S
Glass Transition region Log G(t) Segmental Relaxation Rouse Modes Rubber Reptation Flow 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 Log t Description of dynamics starts with Langevin equation (balance of forces). Rouse model is the basis for the description of chain dynamics in concentrated solutions and melts, while Zimm model describes dynamics of dilute solutions and semidilute solutions on short time and length scales. Reptation model describes entangled chain dynamics, but requires contour length fluctuations correction. Segmental dynamics involves over-barrier motions (e.g. backbone rotation). Freezing of segmental relaxation is the glass transition. It depends on chain rigidity and side groups bulkiness, and on molecular weight. Its relaxation time shows non- Arrhenius temperature dependence. Mechanical properties of glassy polymers depend on secondary relaxations. The latter seem to define also whether polymer will be ductile or brittle. Rubber elasticity is defined by entropy of the chain deformation and depends on molecular weight between crosslinking. T g (M n ) [K] 380 360 340 320 300 280 260 240 220 Tg(inf)-K/M n Tg(M n ) in PS 10 3 10 4 10 5 10 M 6 n